C. Letellier & R. Gilmore,
Poincaré sections for a new three-dimensional toroidal attractor,
Journal of Physics A, 42, 015101, 2009. Here
Abstract: A new 3D autonomous dynamical system proposed by Li  produces a chaotic attractor whose global topological properties are unusual. The attractor has a rotation symmetry and only a single real fixed point for the parameters used in his study. The symmetric, complex pair of fixed points cannot be ignored: they play a major role in organizing the motion on a surface of peculiar toroidal type. We describe this attractor, propose a simple, intuitive model to understand it, show that it is of toroidal type and of genus three, construct a global Poincar´e surface of section with two disjoint components and use this section to locate unstable periodic orbits and determine their topological period. We also show that its image attractor is of genus one and supports flow on a simple wrinkled torus. Finally, we use the interplay between the original covering attractor and its image as an aid to understand why the Li attractor is of genus-three type.
U. S. Freitas, C. Letellier & L. A. Aguirre
Failure for distinguishing colored noise from chaos by the ``Noise titration’’ technique
Physical Review E, 79, 035201R, 2009.
Abstract: Identifying chaos in experimental data - noisy data - remains a challenging problem for which conclusive arguments are still very difficult to provide. In order to avoid problems usually encountered with techniques based on geometrical invariants (dimensions, Lyapunov exponent, etc.), Poon and Barahona introduced a numerical titration procedure which compares one-step-ahead predictions of linear and nonlinear models . We investigate the aformentioned technique in the context of colored noise or other types of nonchaotic behaviors. The main conclusion is that in several examples noise titration fails to distinguish such nonchaotic signals from low-dimensional deterministic chaos.
C. Letellier, D. Amroun & G. Martel
Intermittencies on tori: a way to characterize them
Chaos, Solitons & Fractals, 39, 479-485, 2009.
Abstract: Since the three types of intermittency have been theoretically described, many experimental observations of such regimes have been reported. Chaotic behaviors occurring after torus breakdowns and quasi-periodic regimes are also very often observed. It is not so surprising that intermittencies on tori were never reported as soon as it is understood that these common characteristic of intermittencies should be investigated in a Poincare´ section of a Poincare´ section, that is, in a set which is not possible to define. A specific approach is therefore required to identify them as shown in the paper with two examples of type-I intermittency on tori solution to two different systems.
C. Letellier, L. A. Aguirre & U. S. Freitas
Frequently asked questions about global modeling
Chaos, 19, 023103, 2009.
Abstract: When a global model is attempted from experimental data, some preprocessing might be required. Therefore it is only natural to wonder what kind of effects the preprocessing might have on the modeling procedure. This concern is manifested in the form of recurrent frequently asked questions, such as “how does the preprocessing affect the underlying dynamics?” This paper aims at providing answers to important questions related to i) data interpolation, ii) data smoothing, iii) data estimated derivatives, iv) model structure selection, and v) model validation. The answers provided will hopefully remove some of those doubts and one shall be more confident not only on global modeling but also on various data analyses which may be also dependent on data preprocessing.
L. A. Aguirre & C. Letellier
Modeling Nonlinear Dynamics and Chaos: A Review
Mathematical Problems in Engineering, 2009, 238960, 2009. Here.
Abstract: This paper reviews the major developments of modeling techniques applied to nonlinear dynamics and chaos. Model representations, parameter estimation techniques, data requirements, and model validation are some of the key topics that are covered in this paper, which surveys slightly over two decades since the pioneering papers on the subject appeared in the literature.
C. Letellier & L. A. Aguirre
Symbolic observability coefficients for univariate and multivariate analysis
Physical Review E, 79, 066210, 2009. Here
Abstract: In practical problems, the observability of a system not only depends on the choice of observable(s) but also on the space which is reconstructed. In fact starting from a given set of observables, the reconstructed space is not unique, since the dimension can be varied and, in the case of multivariate measurement functions, there are various ways to combine the measured observables. Using a graphical approach recently introduced, we analytically compute symbolic observability coefficients which allow to choose from the system equations the best observable, in the case of scalar reconstructions, and the best way to combine the observables in the case of multivariate reconstructions. It is shown how the proposed coefficients are also helpful for analysis in higher dimension.
J.-M. Ginoux & C. Letellier
Flow curvature manifolds for shaping chaotic attractors: I. Rössler-like systems
Journal of Physics A, 42, 285101, 2009. Here
Abstract: Poincaré recognized that phase portraits are mainly structured around fixed points. Nevertheless, the knowledge of fixed points and their properties is not sufficient to determine the whole structure of chaotic attractors. In order to understand how chaotic attractors are shaped by singular sets of the differential equations governing the dynamics, flow curvature manifolds are computed. We show that the time-dependent components of such manifolds structure Rösslerlike chaotic attractors and may explain some limitation in the development of chaotic regimes.
U. Freitas, E. Roulin, J.-F. Muir & C. Letellier
Identifying chaos from heart rate: The right task?
Chaos, 19, 028505, 2009. Here
Abstract: Providing a conclusive answer to the question “is this dynamics chaotic?” remains very challenging when experimental data are investigated. We showed that such a task is actually a difficult problem in the case of heart rates. Nevertheless, an appropriate dynamical analysis can discriminate healthy subjects from patients.
C. Letellier & J.-M. Ginoux
Development of the nonlinear dynamical systems theory from radio-engineering to electronics
International Journal of Bifurcation & Chaos, 19 (7), 2131–2163, 2009.
Abstract: Although initial results that contributed to the emergence of the nonlinear dynamical system theory arose from astronomy (the three-body problem), many subsequent developments were related to radio engineering and electronics. The path between the van der Pol equation and the Chua circuit is thus reviewed through main historical contributions.
 Physics Letters A, 372, 387, 2008
 Proceedings of the National Academy of Sciences (USA), 98, 7107, 2001